On Berndt-Straub conjectures about zeros of generalized Ramanujan polynomials

Abstract

We settle two conjectures of Berndt and Straub asserting that all nonzero zeros of the generalized Ramanujan polynomials $R_k(x;χ,ψ)$ and $S_{2n}(x;χ,χ)$ lie on the unit circle for real nonprincipal Dirichlet characters. We prove their conjecture for $R_k$. For $S_{2n}$, we prove that at most four nonzero zeros lie off the unit circle, all purely imaginary. If $χ_{163}=\left(\frac{\cdot}{163}\right)$ is the primitive quadratic character modulo $163$, then $S_6(x;χ_{163},χ_{163})$ has four exceptional zeros, showing that this bound is sharp. This disproves the second conjecture and establishes that the `Riemann hypothesis for period polynomials' can fail for Eisenstein newforms. Further, we introduce a transcendental invariant whose sign determines whether exceptional zeros of $S_{2n}(x;χ,χ)$ occur in all sufficiently large degrees, and show that these zeros converge to transcendental limits off the unit circle. For primitive odd characters, we relate this invariant and these limits to Fekete polynomials.